Inversion - 3-D Seismic Interpretation - page 168

between wells may be easy and reliable, whereas if it is only a few hundred metres
then there will be problems. Often, the range is different in different directions, being
for example greater along strike of a depositional system. Knowing the variogram, a
mathematical methodology called kriging allows us to interpolate between the sparse
well data in a way that honours the spatial variability of the property. In addition,
seismic information can be incorporated by using a methodology called cokriging. For
example, impedance from an inverted dataset might be used as an aid to predicting
reservoir porosity. Cokriging would produce a map that would honour the 'hard' data
at the wells exactly, and would infer values between the wells based on a combination of
kriged interpolation between them and the 'soft' evidence provided by the impedance
data. However, the resulting map is a smooth map of most likely values; it does not
capture the true variability within the reservoir, because as we have seen in chapter 4
the spatial resolution of a seismic dataset is limited; inversion will improve resolution
vertically but not horizontally. Unfortunately for the reservoir engineer, the flow of
fluids in a real reservoir may be strongly influenced by the inhomogeneities that have
been smoothed out of the interpolated model. Stochastic inversion offers a way to
construct a suite of models ('realisations') of the subsurface that are compatible with
the well and seismic evidence and retain the high spatial frequencies.
A way of doing this is outlined by Haas & Dubrule ( 1994 ) . Suppose that we have
a reservoir volume for which we have a reflectivity seismic dataset, and a number of
well penetrations (fig. 6.9) . At each of four wells in the example, we have an acoustic
impedance trace derived from log data. We start by drawing at random a seismic trace
location, where we wish to estimate the impedance trace (shown in green in fig. 6.9 ). A
candidate impedance trace is simulated from the well data; in effect, impedance values
are drawn at random from a population in such a way that the statistics of the spatial
variability of acoustic impedance derived from the well information are honoured. This
means that the candidate trace will contain the short-wavelength vertical variation that is
found in the well logs. From this impedance trace the reflection response is calculated,

Well 1

Well 2

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Well 3

Well 4

Fig. 6.9 Principle of stochastic inversion.